Two solid spherical planets of equal radii R have masses 4 M and 9 M. Their centres are separated by a distance 6 R. A projectile of mass m is sent from the planet of mass 4 M towards the heavier planet. If the distance r of the point from the lighter planet where the gravitational force on the projectile is zero is k R, then calculate k.

Question

Two solid spherical planets of equal radii R have masses 4 M and 9 M. Their centres are separated by a distance 6 R. A projectile of mass m is sent from the planet of mass 4 M towards the heavier planet. If the distance r of the point from the lighter planet where the gravitational force on the projectile is zero is k R, then calculate k. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

The gravitational force on projectile of mass m at a distance r from a planet of mass 4 M is zero. ∴ (G(4 M) m/r2)=(G(9 M) m/(6 R-r)2) or 4(6 R-r)2=9 r2 or 2(6 R-r)=3 r or r=2.4 R

The gravitational force on projectile of mass m at a distance r from a planet of mass 4M is zero. ∴r2G(4M)m​=(6R−r)2G(9M)m​ or 4(6R−r)2=9r2 or 2(6R−r)=3r or r=2.4R

The gravitational force on projectile of mass $m$ at a distance $r$ from a planet of mass $4 M$ is zero.
$∴ \frac{G ( 4 M ) m}{r ^{2}} = \frac{G ( 9 M ) m}{( 6 R - r ) ^{2}}$
or $4 (6 R - r)^{2} = 9 r^{2}$
or $2 (6 R - r) = 3 r$
or $r = 2.4 R$